Electromagnetic Wave Resistivity Tool Having A Tilted Antenna For Geosteering Within A Desired Payzone

ABSTRACT

This invention is directed to a downhole method and apparatus for simultaneously determining the horizontal resistivity, vertical resistivity, and relative dip angle for anisotropic earth formations. The present invention accomplishes this objective by using an antenna configuration in which a transmitter antenna and a receiver antenna are oriented in non-parallel planes such that the vertical resistivity and the relative dip angle are decoupled. Preferably, either the transmitter or the receiver is mounted in a conventional orientation in a first plane that is normal to the tool axis, and the other antenna is mounted in a second plane that is not parallel to the first plane. This invention also relates to a method and apparatus for steering a downhole tool during a drilling operation in order to maintain the borehole within a desired earth formation. The steering capability is enabled by computing the difference or the ratio of the phase-based or amplitude-based responses of the receiver antennas which are mounted in planes that are not parallel to the planes of the transmitter antennas. Although this invention is primarily intended for MWD or LWD applications, this invention is also applicable to wireline and possibly other applications.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.11/198,066, filed on Aug. 5, 2005, which in turn is a continuation ofU.S. application Ser. No. 10/616,429, filed on Jul. 9, 2003, which inturn is a continuation of U.S. application Ser. No. 10/255,048, filed onSep. 25, 2002, which in turn was a continuation of U.S. application Ser.No. 09/615,501, filed on Jul. 13, 2000, which in turn was acontinuation-part of U.S. application Ser. No. 09/238,832 filed Jan. 28,1999, all of which are hereby incorporated by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to a method and apparatus fordetermining the horizontal and vertical resistivities and the relativedip angle of a borehole in an electrically anisotropic earth formation.More specifically, this invention relates to an electromagnetic waveinduction technique for measuring resistivity using a tool having anantenna that is disposed within a plane which is inclined with respectto a plane that is normal to the axis of the tool. This invention alsorelates to a method and apparatus for steering a downhole tool during adrilling operation in order to maintain the borehole within a desiredearth formation.

2. Description of the Related Art

In the field of petroleum well drilling and logging, electromagneticwave resistivity sensors are frequently used to provide an indication ofthe electrical resistivity of rock formations surrounding an earthborehole. Such information regarding resistivity is useful inascertaining the presence or absence of hydrocarbons. A typicalelectromagnetic wave resistivity tool comprises a transmitter antennaand a pair of receiver antennas located at different distances from thetransmitter antenna along the axis of the tool. The transmitter antennais used to transmit electromagnetic waves into the surroundingformation. In turn, the magnetic field in the formation induces anelectrical voltage in each receiver antenna. Due to geometric spreadingand absorption by the surrounding earth formation, the induced voltagesin the two receiving antennas have different phases and amplitudes.Experiments have shown that the phase difference (Φ) and amplitude ratio(attenuation, A) of the induced voltages in the receiver antennas areindicative of the resistivity of the formation. The point in theformation (as defined by a radial distance from the tool axis) to whichsuch a resistivity measurement pertains is a function of the frequencyof the transmitter and the distance from the transmitter to themid-point between the two receivers. Thus, one may achieve multipleradial depths of investigation of resistivity either by providingmultiple transmitters at different distances from the receiver pair orby operating a single transmitter at multiple frequencies.

If a formation is electrically isotropic, the resistivities measured atthe various depths of investigation by such an electromagnetic waveresistivity tool will be the same. However, if the resistivitiescorresponding to the various depths of investigation are different, suchdifferences indicate that the formation being measured is electricallyanisotropic. In electrically anisotropic formations, the anisotropy isgenerally attributable to extremely fine layering during the sedimentarybuild-up of the formation. Hence, in a formation coordinate systemoriented such that the x-y plane is parallel to the formation layers andthe z axis is perpendicular to the formation layers, resistivities R_(x)and R_(y) in directions x and y, respectively, are the same, butresistivity R_(z) in the z direction is different from R_(x) and R_(y).Thus, the resistivity in a direction parallel to the plane of theformation (i.e., the x-y plane) is known as the horizontal resistivity,R_(h), and the resistivity in the direction perpendicular to the planeof the formation (i.e., the z direction) is known as the verticalresistivity, R_(v). The index of anisotropy, α, is defined asα=[R_(v)/R_(h)]^(1/2).

The relative dip angle, θ, is the angle between the borehole axis (toolaxis) and the normal to the plane of the formation. If the axis of anelectromagnetic wave resistivity tool is perpendicular to the plane ofan anisotropic formation (i.e., θ=0°), both the phase shift andamplitude attenuation measurements reflect only the horizontalresistivity. However, if the axis of the tool is inclined with respectto the normal of the formation plane (i.e., for non-zero relative dipangle), the rock anisotropy affects the resistivity derived from phaseshift measurements (“phase shift resistivity” or R_(Φ)) differently thanit affects the resistivity derived from amplitude attenuationmeasurements (“amplitude attenuation resistivity” or R_(A)). For smallrelative dip angles (e.g., θ less than about 45°), the differencebetween phase shift and amplitude attenuation resistivities isrelatively small. However, this difference becomes significant forrelative dip angles greater than about 50°, and the difference is largefor horizontal boreholes (i.e., θ=90°).

Before the present invention, practitioners in the art have used anumber of techniques to determine the anisotropy of earth formations,most of which involve the use of coil antennas to measure resistivity.However, each of the existing techniques suffers from one or moredisadvantages. For example, U.S. Pat. No. 4,980,643, issued Dec. 25,1990 to Gianzero and Su, teaches the use of additional receiver coilsoriented differently from (not co-axial with) conventional receivercoils to detect skew signals induced by skewness in the magnetic fieldpattern in the presence of asymmetrical formations. The axis of suchadditional receiver coils according to the '643 patent is preferablyorthogonal to that of the conventional receiver coils, which areco-axial with the tool axis. However, such orthogonal coil (antenna)configurations are not generally considered practical for a measuringwhile drilling (MWD) or logging while drilling (LWD) tool because, ifthe coil is located in the interior of the tool, the presence of thecoil necessitates a non-conventional mud flow path and decreases themechanical strength of the tool. If, on the other hand, the additionalcoil is located on the exterior of the tool, the coil is susceptible todamage by formation cuttings in the return mud flow.

Similarly, U.S. Pat. No. 5,329,448, issued Jul. 12, 1994 to Rosthal,discloses a method and apparatus for determining the horizontal andvertical resistivities of an earth formation using an iterative errorminimization technique. However, the '448 patent, which discloses aconventional antenna configuration in which the transmitter and receivercoils are aligned co-axially with the tool, does not provide a way todetermine the relative dip angle. Instead, the relative dip angle mustbe obtained from another measurement source or from prior knowledge ofthe formation. Likewise, U.S. Pat. No. 5,656,930, issued Aug. 12, 1997to Hagiwara, discloses a method for determining the anisotropicproperties of subterranean formations comprising thinly laminatedsand/shale sequences using an induction logging tool. However, like the'448 patent, the method of the '930 patent requires the derivation ofthe relative dip angle from another measurement.

In a technical paper entitled “A New Method to Determine HorizontalResistivity in Anisotropic Formations Without Prior Knowledge ofRelative Dip,” 37th SPWLA 37th Annual Logging Symposium, New Orleans,Jun. 16-19, 1996, Hagiwara discloses a method to determine thehorizontal resistivity for deviated boreholes or dipping formationsusing two conventional induction-type resistivity measurements. However,Hagiwara's method does not provide the relative dip angle. To obtain therelative dip angle, the formation anisotropy must be known. Moreover,Hagiwara showed that, for conventional induction logging tools (in whichthe transmitter and receiver antennas are oriented co-axially with thetool), it is impossible to obtain all three parameters (horizontalresistivity, vertical resistivity, and relative dip angle)simultaneously. The reason such a simultaneous solution is not possibleusing conventional induction logging tools is that, in the response ofsuch tools, the vertical resistivity and the relative dip angle arecoupled (i.e., they are not independent).

In European Patent Application No. 97118854.5 by Wu, published May 6,1998, Wu discloses a method and apparatus for determining horizontalconductivity, vertical conductivity, and relative dip angle during adrilling operation. If the relative dip angle is unknown, Wu's techniqueinvolves the formulation of a relationship between the dielectricconstants of the formation to the anisotropic conductivities of theformation. However, in the proof by Hagiwara mentioned above, thedielectric constants are assumed quantities, and their contribution tothe phase shift resistivity is minimal. Therefore, even if thedielectric constants are known, the vertical resistivity and therelative dip angle are still coupled and do not allow for a simultaneoussolution.

It would, therefore, be a significant advancement in the art to providean improved method and apparatus for simultaneously determining thehorizontal resistivity, vertical resistivity, and relative dip angle ina MWD or LWD mode of operation.

Furthermore, to maximize the production of hydrocarbons from a petroleumwell, it would be beneficial to steer the drilling apparatus withrespect to geological bed boundaries in order to maintain the boreholewithin a desired hydrocarbon-bearing earth formation or “payzone.” Asdiscussed by Luling in U.S. Pat. No. 5,241,273, when conventionalresistivity tools traverse geological bed boundaries between formationshaving different resistivities, the responses of such tools exhibithorns at the bed boundaries. Before the advent of the method disclosedby Luling, practitioners in the art considered such horns to beunfortunate anomalies and sought ways to eliminate the horns. Bycontrast, Luling recognized the value of such horns and disclosed amethod of utilizing the horns to assist with directional drilling.However, Luling teaches the use of conventional transmitter and receiverantennas oriented in planes orthogonal to the tool axis, and suchconventional resistivity tools produce the same type of horn responseregardless of whether the tool travels from a region of lowerresistivity to higher resistivity or from a region of higher resistivityto lower resistivity. Consequently, as a conventional resistivity toolapproaches a bed boundary between a first bed having one resistivity anda second bed having a different resistivity, Luling's method indicatesonly the presence of the boundary; Luling's method does not provide anearly indication of whether the resistivity of the second bed is higheror lower than that of the first bed. Only after the tool crosses theboundary and travels a sufficient distance into the second bed willLuling's method provide an indication as to whether the resistivityincreased or decreased so that the driller can make a decision regardingwhich bed is more desirable. As a result, Luling's method may lead tounnecessary penetrations into undesirable beds.

In U.S. Pat. No. 5,230,386, Wu and Wisler disclose another method ofmaintaining a drillstring in a certain formation during a directionaldrilling operation using an electromagnetic propagation resistivitysensor. However, the method of Wu and Wisler requires a referenceresistivity log with which to compare the readings of the resistivitysensor in the directional drilling operation. The reference resistivitylog is obtained by drilling and logging an offset vertical well near thelocation of the desired directional well or by creating an assumed logbased on known geological information concerning the area of interest.The method of Wu and Wisler involves significant drawbacks in terms ofthe time, expense, and uncertainty associated with such a referenceresistivity log.

A technical paper entitled “A New Generation of Electrode ResistivityMeasurements for Formation Evaluation While Drilling,” by Bonner et al.,SPWLA 35th Annual Logging Symposium, Jun. 19-22, 1994, discloses afocused electrode resistivity tool that is azimuthally sensitive andcould be used to steer a drillstring during a directional drillingoperation. However, such electrode resistivity tools operate by forcingelectric current into an earth formation by direct conduction, whichrequires conductive mud. Consequently, such electrode resistivity toolswill not function properly in oil-based mud or in the presence ofresistive invasion.

U.S. Pat. No. 5,508,616 to Sato et al. discloses an induction loggingmethod and apparatus for determining the electric conductivitydistribution of an earth formation in the circumferential directionabout the borehole. The apparatus of the '616 patent employs transmitterand receiver coils that are disposed along the tool axis in a mannersuch that the coils face each other in an inclined fashion at symmetricangles. Although the '616 apparatus provides directional sensitivitywhich may be helpful in steering a drilling tool, the '616 patent doesnot teach how to utilize a transmitter and a receiver oriented atarbitrary inclination angles with respect to each other in order to takeadvantage of the special characteristics of the horns in the resultingresponse.

In light of the foregoing limitations, it would also be a significantadvancement in the art to provide an improved method and apparatus forsteering a downhole tool during a directional drilling operation inorder to maintain the borehole within a desired earth formation byproviding an advance indication of the resistivity of a given formationbefore entry into the formation. Such an improved method and apparatuswould not require an offset vertical well or an assumed referenceresistivity log and would be functional in oil-based or water-based mudand conditions of resistive or conductive invasion.

SUMMARY OF THE INVENTION

Accordingly, this invention is directed to an improved downhole methodand apparatus for simultaneously determining the horizontal resistivity,vertical resistivity, and relative dip angle for anisotropic earthformations. The present invention accomplishes this objective by usingan antenna configuration in which a transmitter antenna and a receiverantenna are oriented in non-parallel planes such that the verticalresistivity and the relative dip angle are decoupled. Preferably, eitherthe transmitter or the receiver is mounted in a conventional orientationin a first plane that is normal to the tool axis, and the other antennais mounted in a second plane that is not parallel to the first plane.Although this invention is primarily intended for MWD or LWDapplications, this invention is also applicable to wireline and possiblyother applications.

This invention is also directed to an improved downhole method andapparatus for steering a downhole tool during a directional drillingoperation in order to maintain the borehole within a desired geologicalformation by providing an advance indication of the resistivity of agiven bed before entry into that bed. Such steering capability isachieved by providing transmitter and receiver antennas that are mountedin non-parallel planes and computing the difference or the ratio of thephase-based or amplitude-based responses of the receiver antennas. Withsuch an antenna arrangement, the ratio or difference of the responsesindicates whether the resistivity of an approaching bed is higher orlower than the resistivity of the present bed. With such information,the driller may steer the drilling apparatus in order to maintain theborehole in a desired geological bed.

BRIEF DESCRIPTION OF THE DRAWINGS

This invention may best be understood by reference to the followingdrawings:

FIG. 1 is a schematic elevational view of an electromagnetic waveresistivity tool in accordance with the present invention.

FIG. 2 is a schematic elevational view of a portion of the tool of FIG.1 showing a block diagram of the circuitry used in accordance with thepresent invention.

FIG. 3 is a schematic pulse diagram illustrating a time sequence oftransmitter pulses of the tool of FIG. 1.

FIG. 4 is a schematic pulse diagram illustrating an alternative timesequence of transmitter pulses of the tool of FIG. 1.

FIG. 5 is a schematic diagram illustrating the antenna geometry of aconventional electromagnetic wave resistivity tool having a transmitterantenna and a receiver antenna, both of which are mounted to the tool ina plane that is orthogonal to the axis of the tool.

FIG. 6 is a schematic diagram illustrating the antenna geometry of anelectromagnetic wave resistivity tool in accordance with the presentinvention having a transmitter antenna mounted to the tool in a planethat is orthogonal to the axis of the tool and a receiver antennamounted to the tool in a plane that is not orthogonal to the axis of thetool.

FIG. 7 is a schematic diagram illustrating several possibletransmitter/receiver antenna configurations in accordance with thepresent invention.

FIG. 8 is a schematic block diagram illustrating the process of solvingfor horizontal conductivity, vertical conductivity, and relative dipangle in accordance with the present invention.

FIG. 9 is a schematic perspective view of a Cartesian coordinate systemin a sedimentary earth formation.

FIG. 10 is a graph of amplitude attenuation versus resistivity for atypical earth formation.

FIG. 11 is a graph of phase shift versus resistivity for a typical earthformation.

FIG. 12 is a graph of apparent resistivity versus relative dip angle fora typical earth formation using a single transmitter-receiver pair.

FIG. 13 is a graph of apparent resistivity versus relative dip angle fora typical earth formation using three transmitter-receiver pairs at asingle frequency.

FIG. 14 is a graph of apparent resistivity versus relative dip angle fora typical earth formation using a single transmitter-receiver pair atthree different frequencies.

FIG. 15 is a schematic diagram showing a typical antenna arrangement fora conventional resistivity tool.

FIG. 16 is a schematic diagram showing one possible antenna arrangementfor a tool in accordance with the present invention.

FIG. 17 is a schematic diagram showing another possible antennaarrangement for a tool in accordance with the present invention.

FIG. 18 is a schematic diagram showing yet another possible antennaarrangement for a tool in accordance with the present invention.

FIG. 19 is a schematic diagram showing yet another possible antennaarrangement for a tool in accordance with the present invention.

FIG. 20 is a schematic diagram showing yet another possible antennaarrangement for a tool in accordance with the present invention.

FIG. 21 is a graph of borehole measured depth versus true vertical depthfor a borehole in a hypothetical earth formation.

FIG. 22 is a graph showing the resistivity response of the conventionaltool of FIG. 15 as it traverses through the borehole of FIG. 21.

FIG. 23 is a graph showing the resistivity response associated withtransmitter T₁ of the tool of FIG. 19 as it traverses through theborehole of FIG. 21.

FIG. 24 is a graph showing the resistivity response associated withtransmitter T₂ of the tool of FIG. 19 as it traverses through theborehole of FIG. 21.

FIG. 25 is a graph showing the differences of the phase and amplituderesponses associated with FIGS. 23 and 24.

FIG. 26 is a graph showing the differences of the resistivity responsesof FIGS. 23 and 24.

FIG. 27 is a graph showing the ratios of the phase and amplituderesponses associated with FIGS. 23 and 24.

FIG. 28 is a graph showing the ratios of the resistivity responses ofFIGS. 23 and 24.

FIG. 29 is a schematic diagram illustrating a resistivity toolapproaching a boundary between two beds.

FIG. 30 is a schematic diagram showing another alternative antennaarrangement for a tool in accordance with the present invention.

FIG. 31 is a schematic diagram showing yet another alternative antennaarrangement for a tool in accordance with the present invention.

FIG. 32 is a schematic diagram showing still another alternative antennaarrangement for a tool in accordance with the present invention.

FIG. 33 is a schematic diagram showing still another alternative antennaarrangement for a tool in accordance with the present invention.

FIG. 34 is a graph showing the phase shift and amplitude attenuationresponse of receiver R₁ of the tool of FIG. 30 as it traverses throughthe borehole of FIG. 21.

FIG. 35 is a graph showing the phase shift and amplitude attenuationresponse of receiver R₂ of the tool of FIG. 31 as it traverses throughthe borehole of FIG. 21.

FIG. 36 is a graph showing the differences of the phase and amplituderesponses associated with FIGS. 34 and 35.

FIG. 37 is a graph showing the ratios of the phase and amplituderesponses associated with FIGS. 34 and 35.

DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT

FIG. 1 illustrates a logging tool 10 in accordance with the presentinvention suspended in a borehole 12 within an earth formation 13 on astring of drill pipe 14. Drill string 14 includes one or more drillcollars 11. Electromagnetic transmitters (antennas) 16, 18, and 20(sometimes referred to herein as transmitters T₁, T₂, and T₃,respectively) are spaced along the length of logging tool 10 fromelectromagnetic receivers (antennas) 22 and 24 (sometimes referred toherein as R₁ and R₂, respectively). Preferably, transmitters 16, 18, 20and receivers 22, 24 are mounted in recesses in tool 10 (as indicated inFIG. 2) and are covered with a non-conductive material (not shown), asis well known in the art. If a transmitter is designed to operate atmore than one frequency, for example, f₁=2 MHz and f₂=1 MHz, eachreceiver may comprise a pair of coils, with one coil tuned to f₁ and onecoil tuned to f₂. Additionally, if desired, each pair of such receivercoils may be located side by side around the periphery of tool 10 or maybe concentrically stacked. Transmitters 16, 18, 20 and receivers 22, 24may be fabricated in accordance with the teachings of U.S. Pat. No.4,940,943, which is assigned to the assignee of the present inventionand is incorporated herein by reference. It should be appreciated thatthe body of tool 10 is preferably made of steel in order to prevent tool10 from becoming a weak link in the drill string 14. Typically, and in amanner well known in the art, one or more drill collars 11 arethreadably connected to the lower end of logging tool 10, and a drillbit (not illustrated) is threadably connected to the lowest drill collar11.

It should be appreciated that logging tool 10 also has the requisiteelectronic circuitry (illustrated in FIG. 2) for processing the signalsreceived by receivers 22, 24 in accordance with the present invention,thereby converting the received signals into a log or another indicationof formation resistivity. It should also be appreciated that theprocessed signals can be recorded within the electronics section of tool10 or may be fed by a conventional telemetry system (not illustrated) tothe surface for concurrent processing and readout at the surface. Atypical telemetry system generates mud pulses that can be detected atthe earth's surface and are indicative of the processed signals.

Referring to FIG. 2, well logging tool 10 is illustrated as having aplurality of transmitters T₁, T₂, T₃ . . . T_(n). Although a preferredembodiment comprises only three such transmitters (T₁, T₂, and T₃),T_(n) is illustrated for purposes of showing that additionaltransmitters may be used, if desired. It should be appreciated that T₁,T₂, T₃ . . . T_(n) are successively further spaced from the receiverpair R₁ and R₂. The distance between the coils used for R₁ and R₂ ispreferably six inches along the longitudinal axis of tool 10, but otherreceiver spacings may also be used. The distance between the receiverpair and the successively spaced transmitters will vary in someapplications, as discussed hereinafter in greater detail. A preferredconfiguration contains a distance between T₁ and R₁/R₂ of 12 inches/18inches; a distance between T₂ and R₁/R₂ of 24 inches/30 inches; and adistance between T₃ and R₁/R₂ of 36 inches/42 inches. In the foregoingsentence, it should be understood that the term “12 inches/18 inches,”for example, indicates that the distance between T₁ and R₁ is 12 inchesand that the distance between T₁ and R₂ is 18 inches, based upon R₁ andR₂ being six inches apart. Such spacing configurations are sometimesreferred to herein using an abbreviated expression of, for example,“12/18.”

Still referring to FIG. 2, a plurality of amplifiers A₁, A₂, A₃ . . .A_(n) are coupled to the transmitter coils T₁, T₂, T₃ . . . T_(n),respectively. The plurality of amplifiers, in turn, are driven,respectively, by oscillators F₁, F₂, F₃ . . . F_(n). The operatingfrequencies of the oscillators are preferably between about 0.5 MHz upto about 4 MHz. Because of power attenuation at greater depths ofinvestigation, such as is the case with the longer spaced transmitters,the frequencies preferably conform to the relationship F₁≧F₂≧F₃≧ . . .F_(n). The oscillators F₁, F₂, F₃ . . . F_(n) are controlled by atransmitter enable circuitry 30, which interfaces with a microprocessor32, which in turn interfaces with a communication interface circuit 34and an analog-to-digital (A/D) converter 36. Communication interfacecircuit 34 is conventional and provides an interface between computers(not shown), an internal memory (not shown), a mud pulser (not shown),microprocessor 32, and operators or computers at the earth's surface(not shown) after the tool 10 is removed to the earth's surface.

The differential receivers R₁ and R₂ are respectively connected toamplifiers 40 and 42, which are connected, respectively, to mixercircuits 44 and 46. Oscillators F₁′, F₂′, F₃′ . . . F_(n)′ are coupledto an oscillator select circuit 48, the output of which is connected tothe respective inputs of mixer circuits 44 and 46. Oscillator selectcircuit 48 receives its inputs from microprocessor 32.

The respective outputs of mixer circuits 44 and 46 drive low passfilters 50 and 52, respectively, the outputs of which drive amplitudemeasurement circuits 54 and 56, respectively. The outputs of amplitudemeasurement circuits 54 and 56 are connected to a multiplexer circuit60. The outputs of low pass filter circuits 50 and 52 are also connectedto the inputs of a relative phase measurement circuit 62, the output ofwhich is fed into multiplexer 60.

In the operation of the device and circuitry illustrated in FIG. 2, itshould be appreciated that it is desirable to process intermediatefrequency signals rather than the nominal signals received by receiverpair R₁ and R₂. Accordingly, the oscillators F₁′, F₂′, F₃′ . . . F_(n)′are selected to be very near the corresponding frequencies F₁, F₂, F₃ .. . F_(n). For example, F_(n) can be set at a frequency of 1.998 MHz andthus provide an intermediate frequency coming out of mixer circuit 44 or46 of 0.002 MHz (2 KHz). In a similar fashion, F₂′ and F₃′ can be set at1.998 MHz and 0.998 MHz, respectively. Thus, the only signals that passto low pass filters 50 and 52 will be the intermediate frequencies whichare obtained by mixing the frequencies of F₁, F₂, F₃ . . . F_(n) withthe frequencies F₁′, F₂′, F₃′ . . . F_(n)′, respectively. It should beappreciated that amplitude measurement circuit 54 provides a measure ofthe amplitude of the signal received by receiver R₁, whereas amplitudemeasurement circuit 56 measures the amplitude of the incoming signalsreceived by receiver R₂. Similarly, relative phase measurement circuit62 provides an indication of the phase difference between the signalsreceived at receiver R₁ and the signals received at receiver R₂. As iswell known in the art, the amplitude measurements (ratios, A) and therelative phase measurements (Φ) are both indicative of formationresistivity. Such measurements may be used to generate plots such asthose shown in FIGS. 12-14 for a typical earth formation having ahorizontal resistivity of 1 ohm-m and a vertical resistivity of 4 ohm-m.FIG. 12 depicts amplitude attenuation resistivity and phase shiftresistivity as a function of relative dip angle using a singletransmitter-receiver pair at a single frequency. FIG. 13 depicts phaseshift resistivity as a function of relative dip angle using threetransmitter-receiver pairs at a single frequency. FIG. 14 depicts phaseshift resistivity as a function of relative dip angle using a singletransmitter-receiver pair at three different frequencies.

It should also be appreciated that the frequencies F₁, F₂, F₃ . . .F_(n) could all be the same frequency except for the practicalconsiderations of power loss in the formation due to the increaseddistance the signals have to travel through the formation. However, theconventional multiplexer circuitry 60 used with this system enables timeseparation between the sequential pulsing of the transmitters T₁, T₂, T₃. . . T_(n). For example, as illustrated in FIG. 3, transmitter T₁ canbe pulsed for one second, followed by no pulse for one second, followedby the pulsation of transmitter T₂ for one second, followed by no pulsefor one second, followed by a pulsing of transmitter T₃ for one second,and so on. Quite obviously, the duration of the pulsing for eachtransmitter can be varied, as well as the duration of no pulsing inbetween, for example, as illustrated in FIG. 4. It should be appreciatedthat the expression “time separation” between pulses includes thepreferred embodiment of having one pulse commence immediately with thetermination of the immediately preceding pulse. As desired, the durationof the pulses controlling T₁ may vary from the duration of the pulsesfor T₂, which may vary from the duration of the pulses for transmitterT₃, and so on, in order to provide a signature of the received pulses atreceivers R₁ and R₂ to better identify the transmitters and thus thedepth of investigation for the particular pulses being received. Thus,measurements are made to different depths into the formation byactivating each transmitter at a different time such that only onetransmitter is active at any one time and by recording or telemeteringthe received phase difference and/or amplitudes (amplitude ratio)corresponding to each transmitted signal. Alternatively, thetransmitters T₁, T₂, T₃ . . . T_(n) could all be operated at differentfrequencies and could be pulsed simultaneously, with the separation ofsignals being a function of frequency difference rather than timeseparation in accordance with a preferred embodiment of this invention.However, those skilled in the art will recognize that simultaneoustransmission of all of the transmitter signals will usually requireadditional filters and processing circuitry to enable the instrument toproperly discriminate between the different frequencies.

As discussed above, due to the nature of sedimentary formations,practitioners in the art use the term “horizontal” to denote the planeof the formation (i.e., the x-y plane of FIG. 9), and practitioners usethe term “vertical” to denote the direction perpendicular to the planeof the formation (i.e., the z direction of FIG. 9, which is thedirection of sedimentary build-up). For convenience in distinguishingbetween these terms of art and the ordinary directions associated withthe earth's gravity, FIGS. 5 and 6 utilize the following terms: “truevertical” indicates the direction of the earth's gravity; “truehorizontal” indicates the direction perpendicular to the earth'sgravity; “formation vertical” indicates the direction perpendicular tothe plane of the formation; and “formation horizontal” indicates theplane of the formation. In this description, the terms “horizontal” and“vertical” are intended to have the meanings associated with “formationhorizontal” and “formation vertical,” respectively. In FIGS. 5 and 6, δis the hole deviation angle (the angle between the borehole/tool axisand the true vertical), and ψ is the bed dip angle (the angle betweenthe formation bed plane and the true horizontal).

Referring to FIG. 5, a transmitter coil (antenna) with a magnetic momentM_(T) can be considered as the superposition of a horizontal magneticdipole (HMD) and a vertical magnetic dipole (VMD), with correspondinghorizontal and vertical component magnetic moments M_(T) _(h) and M_(T)_(v) , respectively, which are given by the equationsM_(T) _(h) =M_(T) sin θ=I_(t)A_(t) sin θ  [1]M_(T) _(v) =M_(T) cos θ=I_(t)A_(t) cos θ  [2]where

I_(t)=the current in the transmitter coil,

A_(t)=the cross-sectional area of the transmitter coil, and

θ=the relative dip angle (the angle between the tool axis and the normalto the formation).As shown by Luling, M. G., “Processing and Modeling 2-MHz ResistivityTools in Dipping, Laminated, Anisotropic Formations,” SPWLA 35th AnnualLogging Symposium, Jun. 19-22, 1994, the HMD produces magnetic fieldsH_(hx) and H_(hz), and the VMD produces magnetic fields H_(vx) andH_(vz) as follows: $\begin{matrix}{H_{hx} = {\frac{M_{T}\sin\quad\theta}{4\pi}( {{\frac{{\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L}}{L^{3}}\lbrack {{3\sin^{2}\theta} - 1 + {k_{h}^{2}L^{2}\cos^{2}\theta} + \frac{{\mathbb{i}}\quad k_{h}L}{\sin^{2}\theta} + {i\quad k_{h}L} - {3{\mathbb{i}}\quad k_{h}L\quad\sin^{2}\theta}} \rbrack} - {\frac{i\quad k_{h}}{\sin^{2}\theta}{\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L\quad\beta}}} ){\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L}}} & \lbrack 3\rbrack \\{H_{hz} = {\frac{M_{T}\cos\quad\theta}{4\pi}( {\frac{{\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L}}{L^{3}}\lbrack {{3\cos\quad{\theta sin}\quad\theta} - {k_{h}^{2}L^{2}\cos\quad{\theta sin}\quad\theta} - {3i\quad k_{h}L\quad\cos\quad{\theta sin}\quad\theta}} \rbrack} )}} & \lbrack 4\rbrack \\{H_{vx} = {\frac{M_{T}\sin\quad\theta}{4\pi}( {\frac{{\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L}}{L^{3}}\lbrack {{3\cos\quad{\theta sin}\quad\theta} - {k_{h}^{2}L^{2}\cos\quad{\theta sin}\quad\theta} - {3i\quad k_{h}L\quad\cos\quad{\theta sin}\quad\theta}} \rbrack} )}} & \lbrack 5\rbrack \\{{H_{vz} = {\frac{M_{T}\cos\quad\theta}{4\pi}( {\frac{{\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L}}{L^{3}}\lbrack {{3\cos^{2}\quad\theta} - 1 + {k_{h}^{2}L^{2}\sin^{2}\quad\theta} - {3i\quad k_{h}L\quad\cos^{2}\quad\theta} + {{\mathbb{i}}\quad k_{h}L}} \rbrack} )}}{where}{k_{h} = \sqrt{\omega^{2}{\mu( {ɛ_{h} - \frac{{\mathbb{i}\sigma}_{h}}{\omega}} )}}}{k_{v} = \sqrt{\omega^{2}{\mu( {ɛ_{v} - \frac{{\mathbb{i}\sigma}_{v}}{\omega}} )}}}{\beta = \sqrt{{\cos^{2}\theta} + {( \frac{k_{v}}{k_{h}} )^{2}\sin^{2}\theta}}}} & \lbrack 6\rbrack\end{matrix}$

-   k_(h)=the complex wave number in the horizontal direction-   k_(v)=the complex wave number in the vertical direction-   ω=the angular frequency (in radians/second) of the transmitter    coil=2πf-   f=the frequency of the transmitter coil (in Hertz)-   μ=the magnetic permeability of the formation (assume μ=μ_(air)=1)-   σ_(h)=the horizontal conductivity of the formation-   σ_(v)=the vertical conductivity of the formation-   ε_(h)=the horizontal dielectric constant (assumed)-   ε_(v)=the vertical dielectric constant (assumed)-   L=the distance between the transmitter coil and the receiver coil-   i=√{square root over (−1)}

If a receiver is parallel to the transmitter, for a conventionalconfiguration as shown in FIG. 5 in which ξ_(T)=ξ_(R)=90°, the H_(z)field in the receiver loop is given by the equationH _(z)=(H _(hx) +H _(vx)) sin θ+(H _(vz) +H _(hz)) cos θ  [7]and the induced voltage in the receiver loop isV=iωA_(r)μH_(z)   [8]where A_(r) is the cross-sectional area of the receiver coil.Substituting Eqs. [3], [4], [5], [6], and [7] into Eq. [8] yields$\begin{matrix}{V = {\frac{{\mathbb{i}\omega}\quad A_{r}\mu\quad I_{t}A_{t}}{4\pi\quad L^{3}}( {{\lbrack {2 - {i\quad k_{h}L}} \rbrack{\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L}} - {i\quad k_{h}L\quad{\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L\quad\beta}}} )}} & \lbrack 9\rbrack\end{matrix}$

Equation [9] shows that the induced voltage, V, depends on k_(h) and β.In turn, k_(h) depends on σ_(h); and β depends on σ_(h), σ_(v), and θ.These relationships indicate that σ_(v) and θ are dependent, and thisdependency prevents convergence of a simultaneous solution for σ_(h),σ_(v), and θ, as discussed above.

To break this dependency and enable a solution for σ_(h), σ_(v), and θ,the present inventor discovered that one may tilt either the transmitteror the receiver with respect to the tool axis, as shown in FIG. 6(transmitter not tilted, receiver tilted). For a transmitter/receiverconfiguration as shown in FIG. 6 in which ξ_(T)=90° and ξ_(R)<90°, theH_(z) field in the receiver loop is given by the equationH _(z)=(H _(hx) +H _(vx)) sin θ′+(H _(vz) +H _(hz)) cos θ′  [10]Substitution of Eqs. [3], [4], [5], [6], and [10] into Eq. [8] yields$\begin{matrix}{{V = {\frac{{\mathbb{i}\omega}\quad A_{r}\mu\quad I_{t}A_{t}}{4\pi\quad L^{3}}( {{{+ \lbrack {{2\sin\quad{\theta sin}\quad\theta^{\prime}} + {2\cos\quad{\theta cos}\quad\theta^{\prime}}} \rbrack}{\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L}} - {\lbrack {{2i\quad k_{h}L\quad\sin\quad{\theta sin}\quad\theta^{\prime}} - {2i\quad k_{h}L\quad\cos\quad{\theta cos}\quad\theta^{\prime}}} \rbrack{\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L}} + {i\quad k_{h}L\quad\frac{\sin\quad\theta^{\prime}}{\sin\quad\theta}{\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L}} - {i\quad k_{h}L\frac{\sin\quad\theta^{\prime}}{\sin\quad\theta}{\mathbb{e}}^{{\mathbb{i}}\quad k_{h}L\quad\beta}}} )}}{{{where}\quad\theta^{\prime}} = {\theta + \xi_{R} - {90{^\circ}}}}} & \lbrack 11\rbrack\end{matrix}$

-   ξ_(R)=the angle of tilt of the receiver antenna (i.e., the angle    between the plane of the receiver antenna and the tool axis)-   ξ_(T)=the angle of tilt of the transmitter antenna (i.e., the angle    between the plane of the transmitter antenna and the tool axis).

Equation [11] shows that the induced voltage, V, depends on k_(h), β, θ,and θ′. As long as θ is different from θ′, then β can be calculated fromthree measurements using a multiple spacing or multiple frequencyelectromagnetic wave resistivity tool. By tilting either the receiver orthe transmitter of an electromagnetic wave resistivity sensor (i.e., bymaking θ different from θ′), σ_(v) and θ are decoupled, which enables asolution for σ_(h), σ_(v), and θ as described below. Although the aboveformulation is for an untilted transmitter with a tilted receiver, thetheory of reciprocity provides that the same result also applies to atilted transmitter with an untilted receiver. Indeed, both thetransmitter and the receiver may be tilted, provided that the respectiveangles of tilt are not the same, i.e., ξ_(T)≠ξ_(R). For the general casein which both the transmitter and the receiver are tilted at arbitraryangles ξ_(T) and ξ_(R), respectively, Eqs. [1] through [11] apply withthe substitution of θ″ for θ where θ″=θ+ξ_(T)−90°. FIG. 7 illustratesseveral possible transmitter/receiver pair combinations in accordancewith the present invention.

FIG. 8 illustrates the process of obtaining the horizontal resistivity,vertical resistivity, and relative dip angle of an earth formation inaccordance with the present invention. Although the diagram of FIG. 8shows horizontal and vertical conductivity rather than horizontal andvertical resistivity, persons reasonably skilled in the art understandthat conductivity is the reciprocal of resistivity, and eitherconductivity or resistivity may be used to refer to the essentialelectrical property of an earth formation. The solution process beginswith an initialization of the three parameters for which a solution issought, namely, horizontal conductivity (σ_(h)), vertical conductivity(σ_(v)), and relative dip angle (θ). As shown in FIG. 8, σ_(h) and σ_(v)may conveniently be initialized to the reciprocals of the measuredresistivity values (measured log) as obtained from two of the threetransmitter/receiver combinations according to methods well known in theart. Specifically, for each transmitter/receiver combination, thetransmitter is pulsed and the phase shift (Φ) and amplitude attenuation(A) of the receiver pair are measured. Then, using data such as thatrepresented in FIGS. 10 and 11, the amplitude attenuation resistivityand phase shift resistivity, respectively, are obtained. The phase shiftresistivities provide preferred initial estimates for σ_(h) and σ_(v),but the amplitude attenuation resistivities could also be used.Similarly, a convenient initial estimate for θ is 60°, which is withinthe range in which anisotropy typically becomes evident. Persons skilledin the art will appreciate that these preferred values for the initialparameter estimates serve the purpose of convenience to enhance theconvergence of a solution. However, these particular values are notessential to the success of this invention, and the initial parameterestimates may be arbitrary.

Still referring to FIG. 8, after the parameters are initialized, theseparameters are used to calculate theoretical “induced” voltages, V₁ andV₂, in receivers R₁ and R₂, respectively, for each transmitter accordingto Eq. [11]. Next, the calculated voltages are used to obtain computedresistivities R_(C) ₁ , R_(C) ₂ , and R_(C) ₃ (computed log)corresponding to each transmitter/receiver pair combination,respectively. Again, the computed resistivities are obtained accordingto methods well known in the art using data such as that shown in FIGS.10 and 11, and the phase shift resistivities are preferred over theamplitude attenuation resistivities. The computed resistivities are thencompared to the measured resistivities, and the difference between thecomputed resistivities and the measured resistivities is used to form asuitable error measurement. If the magnitude of the error is less thanor equal to an allowable error value, E_(allow), then the current valuesfor σ_(h), σ_(v), and θ are taken to be the solution. Otherwise, thevalues for σ_(h), σ_(v), and θ are incremented in an iterativeoptimization routine until the error is within the allowable errorvalue. Any suitable optimization routine may be used, such as a leastsquares method. A preferred optimization method is theLevenberg-Marquardt method discussed by Tianfei Zhu and Larry D. Brown,“Two-dimensional Velocity Inversion and Synthetic SeismogramComputation,” Geophysics, vol. 52, no. 1, January 1987, p. 37-50, whichis incorporated herein by reference.

Turning now to the geosteering aspect of this invention, FIG. 21illustrates a borehole at a relative dip angle of θ=60° traversingthrough a hypothetical earth formation comprising geological zones(beds) A, B, and C having resistivities 1.0, 10.0, and 1.0,respectively. In the hypothetical formation of FIG. 21, zone B is apayzone. FIG. 15 illustrates a conventional resistivity tool 110 havinga transmitter antenna T₁ and a pair of receiver antennas R₁ and R₂, allof which are arranged orthogonal to the tool axis. As conventionalresistivity tool 110 traverses through the borehole of FIG. 21 withoutrotation, tool 110 produces a response S₁ as shown in FIG. 22, whichincludes horns at the intersection of the borehole with the boundariesof each of zones A, B, and C. Significantly, the horns of response S₁increase in the same direction regardless of whether conventional tool110 is crossing from a zone of lower resistivity to a zone of higherresistivity or vice versa.

By contrast, in accordance with the present invention, FIG. 16 depicts aresistivity tool 120 having a transmitter antenna T₁ that is orthogonalto the tool axis and a pair of receiver antennas R₃ and R₄ that aretilted +45° with respect to the tool axis. As tool 120 traverses throughthe borehole of FIG. 21 without rotation, tool 120 produces a responseS₂ that is the same as response S₄ as shown in FIG. 24. Notably,response S₂ (or S₄) includes horns only at the locations at which tool120 is approaching a zone of lower resistivity. Accordingly, tool 120and the associated response S₂ (or S₄) are referred to herein as a“downward looking tool” and a “downward response.”

Similarly, FIG. 17 depicts a resistivity tool 130 having a transmitterantenna T₁ that is orthogonal to the tool axis and a pair of receiverantennas R₅ and R₆ that are tilted −45° with respect to the tool axis.As tool 130 traverses through the borehole of FIG. 21 without rotation,tool 130 produces a response S₃ as shown in FIG. 23. Notably, responseS₃ includes horns only at the locations at which tool 130 is approachinga zone of higher resistivity. Accordingly, tool 130 and the associatedresponse S₃ are referred to herein as an “upward looking tool” and an“upward response.” Persons reasonably skilled in the art will recognizethat tool 130 is the same as tool 120 rotated 180° about the tool axis.

To obtain both responses S₂ (or S₄) and S₃ from a nonrotating tool, thetool may incorporate receiver antennas R₃, R₄, R₅, and R₆ as shown ontool 140 in FIG. 18. Alternatively, as shown in FIG. 19, the tool 150may include an additional transmitter antenna T₂ on the opposite side ofthe receiver pair from transmitter antenna T₁. Responses S₂ and S₄ arethe same because of the theory of reciprocity. Similarly, tool 160 ofFIG. 20 will yield both responses S₂ and S₅ (which is the same as S₃) ina nonrotating mode of operation. Again, persons reasonably skilled inthe art will recognize that tool 160 is the same as tool 150 rotated180° about the tool axis. If the receiver antennas are mounted inrecesses of the tool as discussed above, tool 150 (or 160) is generallypreferred over tool 140 because tool 150 (or 160) is stronger and easierto manufacture.

In light of the nature of the different responses of the upward anddownward looking antenna configurations of tool 140, 150, or 160, suchtools may be used to steer the drilling apparatus in order to stay in adesired payzone. Specifically, the difference between the upward anddownward looking responses (sometimes referred to herein as the“response difference”) indicates whether the tool is approaching a zoneof higher or lower resistivity than the present zone. For example, FIG.26 is a plot of the difference of responses S₃ and S₄ obtained from tool150 (FIG. 19) as discussed above. As shown in FIG. 26 (and in light ofFIG. 21), if the response difference (S₃−S₄) is increasingly positive,the response difference indicates that the tool is approaching a zone ofhigher resistivity (i.e., about to enter a potential payzone). On theother hand, if the response difference (S₃−S₄) is increasingly negative,the response difference indicates that the tool is approaching a zone oflower resistivity (i.e., about to exit a payzone). Because the spikes(horns) in the plot of the response difference (S₃−S₄) begin to occurbefore the tool actually enters the next bed, the drilling operator hasthe opportunity to change the drilling parameters so that the boreholewill remain in the desired payzone. As shown in FIG. 26, either thephase resistivity or the amplitude resistivity may be used for thispurpose. Alternatively, as indicated in FIG. 25, the raw phase shift oramplitude attenuation responses (i.e., before conversion intoresistivity) may be used rather than resistivity signals. The term“differential signal” is used herein to denote either the raw phaseshift/amplitude attenuation responses or the corresponding resistivitysignals. Comparing FIGS. 25 and 26, it should be noted that the rawphase shift/amplitude attenuation response difference (FIG. 25) spikesin the opposite direction as the resistivity response difference (FIG.26).

Additionally, instead of the response difference, the ratio of theupward and downward looking responses (sometimes referred to herein asthe “response ratio”) may be used. For example, FIG. 27 shows responseratio (S₃/S₄) using the raw phase shift and amplitude attenuationresponses, and FIG. 28 shows response ratio (S₃/S₄) after conversioninto resistivity. Considering FIG. 28, if the resistivity response ratio(S₃/S₄) is increasingly greater than 1.0, the response ratio indicatesthat the tool is approaching a zone of higher resistivity (i.e., aboutto enter a potential payzone); if the resistivity response ratio (S₃/S₄)is increasingly less than 1.0, the response ratio indicates that thetool is approaching a zone of lower resistivity (i.e., about to exit apayzone). Similarly, considering FIG. 27, if the raw phaseshift/amplitude attenuation response ratio (S₃/S₄) is increasingly lessthan 1.0, the response ratio indicates that the tool is approaching azone of higher resistivity (i.e., about to enter a potential payzone);if the raw phase shift/amplitude attenuation response ratio (S₃/S₄) isincreasingly greater than 1.0, the response ratio indicates that thetool is approaching a zone of lower resistivity (i.e., about to exit apayzone). Again, because the spikes (horns) in the plot of the responseratio (S₃/S₄) begin to occur before the tool actually enters the nextbed, the drilling operator has the opportunity to change the drillingparameters so that the borehole will remain in the desired payzone.Generally, the response difference is preferred over the response ratiobecause taking the ratio of the responses normalizes the data andthereby decreases the resolution. The term “output signal” is usedherein to denote either the response difference or the response ratio.

As an alternative to the foregoing configurations which involve a pairof receiver antennas, a configuration involving a single receiverantenna may also be used to steer the drill bit in accordance with thepresent invention. FIG. 30 shows a tool 170 having a single transmitterT₁ orthogonal to the tool axis and a single receiver R₁ tilted +45° withrespect to the tool axis. Instead of measuring the phase difference andamplitude attenuation between a pair of receivers, tool 170 is used tomeasure the phase difference and amplitude attenuation between thetransmitter T₁ and the receiver R₁. In this context, the term“differential signal” is also used herein to denote the raw phase shiftor amplitude attenuation response between a transmitter and a receiverrather than between a pair of receivers. As tool 170 traverses throughthe borehole of FIG. 21 without rotation, tool 170 produces a responseS3 a as shown in FIG. 34. Response S3 a exhibits a spike as tool 170approaches and travels across the boundary from bed A to bed B.Similarly, FIG. 31 shows a tool 180 having a single transmitter T₁orthogonal to the tool axis and a single receiver R₂ tilted −45° withrespect to the tool axis. Again, it will be appreciated that tool 180 isthe same as tool 170 rotated 180 degrees about the tool axis. As tool180 traverses through the borehole of FIG. 21 without rotation, tool 180produces a response S4 a as shown in FIG. 35. Response S4 a exhibits aspike as tool 180 approaches and travels across the boundary from bed Bto bed C. Both responses S3 a and S4 a may be acquired by rotating tool170 or 180 in the borehole, or by using tool 190 as shown in FIG. 32,which has receiver antennas R₁ and R₂ in a cross-wise configuration, orby using tool 200 as shown in FIG. 33, which has two transmitterantennas T₁ and T₂ on either side of a receiver antenna R₁. By takingthe difference (S3 a−S4 a) or the ratio (S3 a/S4 a) of the responses,one may produce the plots shown in FIGS. 36 and 37, respectively, whichmay be used to steer the drill bit. Referring to FIG. 36 in conjunctionwith FIG. 21, as the response difference (S3 a−S4 a) becomes negative,it indicates that the tool is approaching a bed of higher resistivity(i.e., about to enter a potential payzone); conversely, as the responsedifference (S3 a−S4 a) becomes positive, it indicates that the tool isapproaching a bed of lower resistivity (i.e., about to exit a payzone).Similarly, referring to FIG. 37 in conjunction with FIG. 21, as theresponse ratio (S3 a/S4 a) becomes less than 1.0, it indicates that thetool is approaching a bed of higher resistivity (i.e., about to enter apotential payzone); conversely, as the response ratio (S3 a/S4 a)becomes greater than 1.0, it indicates that the tool is approaching abed of lower resistivity (i.e., about to exit a payzone).

Although Luling (U.S. Pat. No. 5,241,273) defined the term “horn” tomean “a sharp local maximum with a peak resistivity at least twice theresistivity on either side of the local maximum,” the present inventiondoes not require such a dramatic change in resistivity for the purposeof geosteering. Rather, as illustrated in FIGS. 25-28 and 35-36, thepresent invention is sensitive to the difference or ratio of theresponses of the downward looking and upward looking antennas. Theability of the present invention to provide an advance indication of abed boundary during horizontal drilling is illustrated in FIG. 29. For atypical resistivity tool, the antennas have a sensitive zone of about3.0 meters radially from the tool axis. Thus, if the tool is drillingthrough a payzone and approaching a bed boundary at an angle of attackof about 2 degrees, the drilling operator will have about 85 meters ofborehole depth in which to steer away from the bed boundary in order tostay in the desired payzone. As with the other examples in thisdisclosure, the foregoing example is for illustrative purposes only andshould not be considered limiting for the present invention.

If a resistivity tool in accordance with the present invention isrotating, the upward and downward responses will vary sinusoidally. In arotating mode of operation, tool 120 of FIG. 16 (or tool 130 of FIG.17), for example, generates both the upward and downward responses.Accordingly, as shown in FIG. 2, a preferred embodiment of thisinvention contains at least one rotational position indicator 70, suchas an accelerometer or magnetometer, which also has a sinusoidal output.By correlating the upward and downward responses with the output of therotational position indicator(s) 70, the orientation of the tool may becalculated according to methods well known in the art. The at least onerotational position indicator 70 preferably comprises magnetometers,inclinometers, or accelerometers; however, other sensor types, such asgyroscopes, may also be used. The function of the at least onerotational position indicator 70 is to provide a sinusoidal electricalsignal as the tool rotates that indicates the rotational orientation ofthe tool with respect to a reference direction, such as the direction ofgravity or magnetic north. A gravitational sensor (such as aninclinometer or accelerometer) does not function satisfactorily when theaxis of the tool is aligned with the gravitational direction (i.e., in avertical orientation), and a magnetic sensor (such as a magnetometer)does not function satisfactorily when the axis of the tool is alignedwith the magnetic north direction. Therefore, the tool preferablycontains at least one gravitational sensor (e.g., an accelerometer) andat least one magnetic sensor (e.g., a magnetometer) so that theorientation of the tool is ascertainable in any position.

More preferably, rotational position indicator 70 may contain both a3-axis fluxgate magnetometer and a 3-axis accelerometer. As is known inthe art, the combination of those two sensor systems enables themeasurement of the toolface, inclination, and azimuth orientation anglesof the borehole. The toolface and hole inclination angles are calculatedfrom the accelerometer sensor output. The magnetometer sensor outputsare used to calculate the hole azimuth. With the toolface, the holeinclination, and the hole azimuth information, a tool in accordance withthe present invention can be used to steer the bit to the desirable bed.Specifically, the response difference or the response ratio can be usedeffectively to enter a desired payzone or to stay within the payzone ofinterest.

The exemplary responses shown in FIGS. 22-28 are for 2-MHz tools with24/30 antenna spacings. However, persons reasonably skilled in the artwill recognize that other frequencies and antenna spacings could beused. Additionally, although the illustrative examples provided hereincomprise antenna tilt angles of +45° and −45°, other tilt angles couldbe used.

Although the foregoing specific details describe a preferred embodimentof this invention, persons reasonably skilled in the art of petroleumwell drilling and logging will recognize that various changes may bemade in the details of the method and apparatus of this inventionwithout departing from the spirit and scope of the invention as definedin the appended claims. Therefore, it should be understood that thisinvention is not to be limited to the specific details shown anddescribed herein.

1. A logging while drilling tool having a tool axis, the toolcomprising: at least one transmitter antenna oriented at a first anglerelative to the tool axis of the logging while drilling tool to transmitan electromagnetic wave into a formation; at least one receiver antennaoriented at a second angle relative to the tool axis to receive anelectromagnetic wave from the formation, wherein the first and secondangles are unequal; and a processor coupled to the at least one receiverantenna to determine at least one formation measurement.
 2. The loggingwhile drilling tool of claim 1, wherein the at least one formationmeasurement comprises at least one of a complex voltage, an amplituderatio, and a phase difference.
 3. The logging while drilling tool ofclaim 2, wherein the formation measurement is determined with referenceto the electromagnetic wave from the transmitter antenna.
 4. The loggingwhile drilling tool of claim 1, wherein the at least one formationmeasurement is a resistivity component of the formation.
 5. The loggingwhile drilling tool of claim 4, wherein the at least one formationmeasurement is vertical resistivity.
 6. The logging while drilling toolof claim 1, wherein the processor calculates vertical and horizontalresistivity of the formation.
 7. The logging while drilling tool ofclaim 6, wherein the processor further calculates a relative dip of theformation.
 8. The logging while drilling tool of claim 1, wherein adifference between the first and second angles is less than 90°.
 9. Thelogging while drilling tool of claim 1, wherein the second angle is 90°.10. The logging while drilling tool of claim 1, further comprising asecond transmitter antenna oriented at the first angle relative to thetool axis to transmit an electromagnetic wave into the formation. 11.The logging while drilling tool of claim 10, further comprising a thirdtransmitter antenna oriented at the first angle relative to the toolaxis to transmit an electromagnetic wave into the formation.
 12. Thelogging while drilling tool of claim 11, wherein the electromagneticsignal sent from the at least one transmitter antenna has a firstfrequency, wherein the electromagnetic signal sent from the secondtransmitter antenna has a second frequency, wherein the electromagneticsignal sent from the third transmitter antenna has a third frequency,and wherein the first, second, and third frequencies are different. 13.The logging while drilling tool of claim 11, wherein the first angle issuch that each of the transmitter antennas are coaxial with the loggingwhile drilling tool, and the second angle is such that the receiverantenna is tilted relative to the tool axis.
 14. The logging whiledrilling tool of claim 1, wherein the transmitter antenna is located ata first position on a tool string and the receiving antenna is locatedat a second, different position on the tool string.
 15. A method thatcomprises: sending an electromagnetic wave from a transmitter antennainto a formation, wherein the transmitter antenna is mounted on alogging while drilling tool at a first angle with respect to a toolaxis; receiving an electromagnetic wave from the formation with areceiver antenna, wherein the receiver antenna is mounted on the loggingwhile drilling tool at a second angle with respect to the tool axis,wherein the first and second angles are different; and determining atleast one resistivity component of the formation based at least in parton the received electromagnetic wave.
 16. The method of claim 15,wherein the resistivity component is vertical resistivity.
 17. Themethod of claim 16, further comprising determining a horizontalresistivity of the formation based at least in part on the receivedelectromagnetic wave.
 18. The method of claim 17, further comprisingdetermining a relative dip angle of the formation based at least in parton the received electromagnetic wave.
 19. The method of claim 15,wherein the at least one resistivity component comprises a phase shift.20. The method of claim 15, wherein the electromagnetic wave from thetransmitter antenna has a first frequency, and wherein the methodfurther comprises: sending a second electromagnetic wave having a secondfrequency from a second transmitter antenna into the formation, whereinthe second frequency is different from the first frequency; sending athird electromagnetic wave having a third frequency from a thirdtransmitter antenna into the formation, wherein the third frequency isdifferent from the first and second frequencies; receivingelectromagnetic waves having the second and third frequencies from theformation with the receiver antenna; and determining at least onecomponent of resistivity of the formation.
 21. The method of claim 20wherein the at least one component of resistivity comprises any one of arelative dip angle of the formation, a horizontal resistivity of theformation, and a vertical resistivity of the formation.
 22. The methodof claim 20, wherein each of the first, second, and third transmitterantennas are coaxial with the logging while drilling tool, and whereinthe receiver antenna is tilted relative to the tool axis.
 23. A boundarydetection tool that comprises: at least one transmitter antenna orientedat a first angle relative to the tool axis to transmit anelectromagnetic wave into a formation; at least one receiver antennaoriented at a second angle relative to the tool axis to receive anelectromagnetic wave from the formation, wherein the first and secondangles are unequal; and a processor coupled to the at least one receiverantenna to determine a bed boundary indicator.
 24. The boundarydetection tool of claim 23, further comprising: a second receiverantenna oriented at the second angle relative to the tool axis, whereinthe processor is also coupled to the second receiver antenna todetermine the bed boundary indicator.
 25. The boundary detection tool ofclaim 24, further comprising: a second transmitter antenna oriented atthe first angle relative to the tool axis, wherein the transmitterantennas are symmetrically spaced around a midpoint between the receiverantennas.
 26. The boundary detection tool of claim 25, wherein the bedboundary indicator is a difference between measurements determined inresponse to operation of the two transmitter antennas.
 27. The boundarydetection tool of claim 23, further comprising a second receiver antennaoriented symmetrically to the at least one receiver antenna, wherein thebed boundary indicator is a difference between measurements determinedfrom the two receiver antennas.
 28. The boundary detection tool of claim23, further comprising a rotational position indicator coupled to theprocessor, wherein the bed boundary indicator is a difference betweenmeasurements determined at opposite azimuthal orientations of the tool.